Crossing the brachistochrone- Faster than light

A way to support the idea of faster than light travel is the following one. In the original description of the problem of wave-particle duality, as we have already mentioned in that section, if we define the momenta of a wave and of a particle, respectively, as

we take an energy of the form

If in this energy equation we set v=c, we take

This energy (which is famous thanks to Einstein) treats the mass m of the system as one and the same.

If however we introduce a mass μ for the wave, then the previous relationships take the form

where the index ‘0,’ in ε₀, refers to the total energy.

This analysis also reveals the aspect that we refer to an energy per wavelength λ. The notion of the brachistochrone makes clearer that the wavelength λ refers to a division of the length L of the brachistochrone, so that, by analogy, the mass μ of the wave per wavelength will refer to a division of the total mass M of the brachistochrone.

This way, while the masses μ and M refer to the oscillations of a wave of spacetime (represented by the brachistochrone), the mass m (commonly referred to as inertial mass) refers to an object travelling on the brachistochrone, whereas the wavelength λ refers to the oscillations of spacetime, perceived as photons (thus it is not the Compton wavelength of the moving object).

Therefore by separating the two masses m and μ, the equation

will imply an object of mass m comparable to the mass μ of the brachistochrone per wavelength, travelling at the speed of light.

The energy E stored in a region of spacetime for all wavelengths is equal to

where M is the mass equivalent of the energy E, according to Einstein’s formula.

This amount of energy can also be seen as the Planck energy of the oscillations of spacetime in the same region, because anything which contains energy vibrates. If λ is the wavelength of such oscillations then the Planck energy related to those oscillations will be

This energy is supposed to be associated with photon radiation, and it is an amount of energy per wavelength λ of the photon, or per photon. The quantity μ refers to the mass equivalent of the energy ε (the energy E per wavelength λ).

If L is the linear dimension of the region of spacetime under concern, and it is composed of a number of N photons of wavelength λ, then the total energy stored in that region will be

where

The main assumption is that an object which moves in the same region of spacetime, causes spacetime to oscillate faster, or the wavelength of the photons by which those oscillations are perceived gets shorter. The increase of the frequency, or the decrease of the wavelength, of the photons is associated with the harmonics n of the oscillations. If λ₀ is the initial wavelength of the photons, and λ_n is the wavelength at some harmonic n, then it will be

The harmonics n, which express how many times the wavelength λ_n of the photons gets shorter with respect to the original wavelength λ₀, and the number N (or N_n), which refers to the amount of photons which comprise the distance L≡L₀ at some harmonic n, as we have already seen, are not necessarily equal to each other:

where N₀ refers to the number of photons which comprise the distance L₀ at the first harmonic (n=1).

Therefore, if we call ε₀ the energy stored in spacetime per the initial wavelength λ₀ of the photons at the first harmonic, n=1,

and if we call ε_n the energy per the wavelength λ_n of the photons at some harmonic n,

then the total energy Ε₀, corresponding to the first harmonic, will be

while the total energy Ε_n, corresponding to the harmonic n, will be

where

Such is the relationship between the wavelength λ_n of the photons and the distance L₀, or between the mass μ_n per wavelength λ_n and the total mass M_n, referring to a region of spacetime.

On the other hand, we have the moving object in the same region of spacetime. If this object’s mass is m, and its speed is v, then its kinetic energy E_k will commonly be

Since it was assumed that the motion of the object disturbs the oscillating spacetime, and that this disturbance leads to the change of the wavelength of the oscillations of spacetime, we can relate the kinetic energy E_k of the moving object to the energy of the oscillations, which we may call E_M,

The factor of ½ in front of the kinetic energy can be dropped assuming, for example, that the total kinetic energy is twice the average kinetic energy. As long as we refer to the total energy, the two forms of energy can be equated, in the sense that one form transforms into the other form, so that, using for the total energy the same symbol E, it will be

A first indication that the speed of the object can be greater than the speed of light is that, logically, the mass M stored in a region of spacetime will be larger than the mass m of the object moving in the same region,

A second remark is that the wavelength λ of the oscillations of spacetime, perceived in the form of photons by an observer on board the moving object, will contract,

The greater the speed of the object is, the smaller the wavelength of the photons will become. Such an aspect can also be seen as a consequence of the fact that the object takes energy from spacetime, which transforms into its own kinetic energy.

Considering the harmonic n, or the number N, of the photons, the general relationship between the mass m of the object (which we may also call m₀, with reference to the inertial mass) and the mass M (or the mass μ per wavelength) stored in spacetime, can be taken as follows,

where

so that if, for example, we assume that the inertial mass m₀ of the object is comparable to the mass M₀ of spacetime at the first harmonic (n=1), then for the speed of the object we have that

If we additionally suppose that the distance L₀ is comparable to the photon’s initial wavelength λ₀, then it will also be

Significantly, in any case, the total energy E_n can be expressed solely in relation to the harmonic n of the photons, and it will be n times squared the energy E₀ of the first harmonic (n=1):

where the condition

is sufficient so that the final speed of the object at any harmonic n can be expressed solely with respect to the number n,

Notes:

Now let’s attempt to put some values in the previous equations. Let’s suppose, for example, that we have a spaceship of mass m comparable to that of a modern supercarrier, so that

Let this spaceship have to travel a distance L of 1ly,

Here we will divide the distance L into Planck lengths l_P (instead of wavelengths λ of some reference photon), so that

This region of space contains a mass equivalent M of N_P Planck masses m_P,

The ratio between the mass M of spacetime and the mass m of the spaceship is

If the spaceship consumes all this amount of available mass M, then its final kinetic energy will be

so that its final speed will be

This is presumably the maximum possible speed for that spaceship.

But what about the energy of the reference photon? Here we will make use of the notion of the cosmic microwave background radiation (CMBR). According to Wikipedia, the photon energy of CMB photons is about 6.627×10⁻⁴eV.
[https://en.wikipedia.org/wiki/Cosmic_microwave_background]

Changing the units into Joules, we have

This energy supposedly will be an energy covering spacetime at the ground state, n=1, and will refer to the energy of the CMB photons per wavelength λ₀, where

The mass equivalent of this energy is

The ratio between the distance L and the wavelength λ₀ will be

Thus the total energy Ε₀ at the first harmonic, or state, will be

corresponding to a mass

The whole energy equation at the first state (n=1) is

In order for the spaceship of mass m to reach the speed of light, it will have to consume an amount of mass equal to M₀:

Presumably, this amount of mass will be sufficient for the spaceship to reach the speed of light.

The value of the wavelength λ of the reference photon which was used here is indicative, and it serves as a standard unit of length in order to divide the total distance L to be traveled.

In any case, the equation for the energy per wavelength,

suggests that an object of mass m, travelling a distance λ at a time τ, can reach a speed v_λ (where here the index ‘λ’ means ‘per wavelength λ’) √μ/m times less than the speed of light, where μ is the mass stored in the distance λ, and which can be used by the moving object. If the object’s mass m is comparable to the mass μ, then the object travelling the distance λ can reach the speed of light at exactly the time τ.

But if the object travels a macroscopic distance L which is N times bigger than the unit distance λ then the equation for the energy stored in that distance,

implies that the object’s final speed v will be √N times greater than its speed v_λ ‘per wavelength λ.’ If the object’s speed v_λ fοr the distance λ is equal to the speed of light, then the object’s total speed for the distance L will be √N times greater than the speed of light.

If we set the unit distance λ equal to Planck length l_P, then the mass μ stored in this distance will be Planck mass m_P. If the object’s mass m is comparable to Planck mass, then that object’s final speed v for the total distance L can be so many times greater than the speed of light, as the square root of the number N_P=L/l_P.

The point is that, given the huge amounts of energy stored in very small regions of spacetime, an object’s final speed can be much greater than the speed of light.